TITLE: Number of $4$-member committees from $3$ women and $5$ men that have at least $2$ women
QUESTION [5 upvotes]: The United States Supreme Court consists of 3 women and 5 men. In how many ways can a 4-member committee be formed if each committee must have at least two women?.
I know that we have $^8C_4=70$ combinations.
I'm stuck on how many committees can be formed with at least two women.
Do I get the combination of committees that include all men and subtract with the $70$?

REPLY [3 votes]: You can select 2 or 3 from the 3 available women and select however many men from the 5 available to complete a committee of 4.
$${^3\mathrm C_2}~{^5\mathrm C_2}+{^3\mathrm C_3}~{^5\mathrm C_1} = 35$$
That is all.